A Projective-to-Conformal Fefferman-Type Construction

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2017
• Main Authors: Hammerl, M., Sagerschnig, K., Šilhan, J., Taghavi-Chabert, A., Zádník, V.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2017
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/149272
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: A Projective-to-Conformal Fefferman-Type Construction / M. Hammerl, K. Sagerschnig, J. Šilhan, A. Taghavi-Chabert, V. Zádník// Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 30 назв. — англ.

We study a Fefferman-type construction based on the inclusion of Lie groups SL(n+1) into Spin(n+1,n+1). The construction associates a split-signature (n,n)-conformal spin structure to a projective structure of dimension n. We prove the existence of a canonical pure twistor spinor and a light-like conformal Killing field on the constructed conformal space. We obtain a complete characterisation of the constructed conformal spaces in terms of these solutions to overdetermined equations and an integrability condition on the Weyl curvature. The Fefferman-type construction presented here can be understood as an alternative approach to study a conformal version of classical Patterson-Walker metrics as discussed in recent works by Dunajski-Tod and by the authors. The present work therefore gives a complete exposition of conformal Patterson-Walker metrics from the viewpoint of parabolic geometry.